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<div class="subTitle">org.apache.commons.math3.linear</div>
<h2 title="Class SingularValueDecomposition" class="title">Class SingularValueDecomposition</h2>
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<pre>public class <span class="strong">SingularValueDecomposition</span>
extends <a href="http://docs.oracle.com/javase/6/docs/api/java/lang/Object.html?is-external=true" title="class or interface in java.lang">Object</a></pre>
<div class="block">Calculates the compact Singular Value Decomposition of a matrix.
 <p>
 The Singular Value Decomposition of matrix A is a set of three matrices: U,
 &Sigma; and V such that A = U &times; &Sigma; &times; V<sup>T</sup>. Let A be
 a m &times; n matrix, then U is a m &times; p orthogonal matrix, &Sigma; is a
 p &times; p diagonal matrix with positive or null elements, V is a p &times;
 n orthogonal matrix (hence V<sup>T</sup> is also orthogonal) where
 p=min(m,n).
 </p>
 <p>This class is similar to the class with similar name from the
 <a href="http://math.nist.gov/javanumerics/jama/">JAMA</a> library, with the
 following changes:</p>
 <ul>
   <li>the <code>norm2</code> method which has been renamed as <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getNorm()"><code>getNorm</code></a>,</li>
   <li>the <code>cond</code> method which has been renamed as <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()"><code>getConditionNumber</code></a>,</li>
   <li>the <code>rank</code> method which has been renamed as <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getRank()"><code>getRank</code></a>,</li>
   <li>a <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()"><code>getUT</code></a> method has been added,</li>
   <li>a <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()"><code>getVT</code></a> method has been added,</li>
   <li>a <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSolver()"><code>getSolver</code></a> method has been added,</li>
   <li>a <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getCovariance(double)"><code>getCovariance</code></a> method has been added.</li>
 </ul></div>
<dl><dt><span class="strong">Since:</span></dt>
  <dd>2.0 (changed to concrete class in 3.0)</dd>
<dt><span class="strong">Version:</span></dt>
  <dd>$Id: SingularValueDecomposition.java 1538368 2013-11-03 13:57:37Z erans $</dd>
<dt><span class="strong">See Also:</span></dt><dd><a href="http://mathworld.wolfram.com/SingularValueDecomposition.html">MathWorld</a>, 
<a href="http://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia</a></dd></dl>
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<td class="colOne"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#SingularValueDecomposition(org.apache.commons.math3.linear.RealMatrix)">SingularValueDecomposition</a></strong>(<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;matrix)</code>
<div class="block">Calculates the compact Singular Value Decomposition of the given matrix.</div>
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<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()">getConditionNumber</a></strong>()</code>
<div class="block">Return the condition number of the matrix.</div>
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<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getCovariance(double)">getCovariance</a></strong>(double&nbsp;minSingularValue)</code>
<div class="block">Returns the n &times; n covariance matrix.</div>
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<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getInverseConditionNumber()">getInverseConditionNumber</a></strong>()</code>
<div class="block">Computes the inverse of the condition number.</div>
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<td class="colFirst"><code>double</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getNorm()">getNorm</a></strong>()</code>
<div class="block">Returns the L<sub>2</sub> norm of the matrix.</div>
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<td class="colFirst"><code>int</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getRank()">getRank</a></strong>()</code>
<div class="block">Return the effective numerical matrix rank.</div>
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<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getS()">getS</a></strong>()</code>
<div class="block">Returns the diagonal matrix &Sigma; of the decomposition.</div>
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<td class="colFirst"><code>double[]</code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSingularValues()">getSingularValues</a></strong>()</code>
<div class="block">Returns the diagonal elements of the matrix &Sigma; of the decomposition.</div>
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<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/DecompositionSolver.html" title="interface in org.apache.commons.math3.linear">DecompositionSolver</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getSolver()">getSolver</a></strong>()</code>
<div class="block">Get a solver for finding the A &times; X = B solution in least square sense.</div>
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<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getU()">getU</a></strong>()</code>
<div class="block">Returns the matrix U of the decomposition.</div>
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<tr class="rowColor">
<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()">getUT</a></strong>()</code>
<div class="block">Returns the transpose of the matrix U of the decomposition.</div>
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<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getV()">getV</a></strong>()</code>
<div class="block">Returns the matrix V of the decomposition.</div>
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<td class="colFirst"><code><a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a></code></td>
<td class="colLast"><code><strong><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()">getVT</a></strong>()</code>
<div class="block">Returns the transpose of the matrix V of the decomposition.</div>
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<pre>public&nbsp;SingularValueDecomposition(<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;matrix)</pre>
<div class="block">Calculates the compact Singular Value Decomposition of the given matrix.</div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>matrix</code> - Matrix to decompose.</dd></dl>
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<h4>getU</h4>
<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getU()</pre>
<div class="block">Returns the matrix U of the decomposition.
 <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the U matrix</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getUT()"><code>getUT()</code></a></dd></dl>
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<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getUT()</pre>
<div class="block">Returns the transpose of the matrix U of the decomposition.
 <p>U is an orthogonal matrix, i.e. its transpose is also its inverse.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the U matrix (or null if decomposed matrix is singular)</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getU()"><code>getU()</code></a></dd></dl>
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<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getS()</pre>
<div class="block">Returns the diagonal matrix &Sigma; of the decomposition.
 <p>&Sigma; is a diagonal matrix. The singular values are provided in
 non-increasing order, for compatibility with Jama.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the &Sigma; matrix</dd></dl>
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<h4>getSingularValues</h4>
<pre>public&nbsp;double[]&nbsp;getSingularValues()</pre>
<div class="block">Returns the diagonal elements of the matrix &Sigma; of the decomposition.
 <p>The singular values are provided in non-increasing order, for
 compatibility with Jama.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the diagonal elements of the &Sigma; matrix</dd></dl>
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<h4>getV</h4>
<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getV()</pre>
<div class="block">Returns the matrix V of the decomposition.
 <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the V matrix (or null if decomposed matrix is singular)</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getVT()"><code>getVT()</code></a></dd></dl>
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<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getVT()</pre>
<div class="block">Returns the transpose of the matrix V of the decomposition.
 <p>V is an orthogonal matrix, i.e. its transpose is also its inverse.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the V matrix (or null if decomposed matrix is singular)</dd><dt><span class="strong">See Also:</span></dt><dd><a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getV()"><code>getV()</code></a></dd></dl>
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<h4>getCovariance</h4>
<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/RealMatrix.html" title="interface in org.apache.commons.math3.linear">RealMatrix</a>&nbsp;getCovariance(double&nbsp;minSingularValue)</pre>
<div class="block">Returns the n &times; n covariance matrix.
 <p>The covariance matrix is V &times; J &times; V<sup>T</sup>
 where J is the diagonal matrix of the inverse of the squares of
 the singular values.</p></div>
<dl><dt><span class="strong">Parameters:</span></dt><dd><code>minSingularValue</code> - value below which singular values are ignored
 (a 0 or negative value implies all singular value will be used)</dd>
<dt><span class="strong">Returns:</span></dt><dd>covariance matrix</dd>
<dt><span class="strong">Throws:</span></dt>
<dd><code><a href="http://docs.oracle.com/javase/6/docs/api/java/lang/IllegalArgumentException.html?is-external=true" title="class or interface in java.lang">IllegalArgumentException</a></code> - if minSingularValue is larger than
 the largest singular value, meaning all singular values are ignored</dd></dl>
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<h4>getNorm</h4>
<pre>public&nbsp;double&nbsp;getNorm()</pre>
<div class="block">Returns the L<sub>2</sub> norm of the matrix.
 <p>The L<sub>2</sub> norm is max(|A &times; u|<sub>2</sub> /
 |u|<sub>2</sub>), where |.|<sub>2</sub> denotes the vectorial 2-norm
 (i.e. the traditional euclidian norm).</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>norm</dd></dl>
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<h4>getConditionNumber</h4>
<pre>public&nbsp;double&nbsp;getConditionNumber()</pre>
<div class="block">Return the condition number of the matrix.</div>
<dl><dt><span class="strong">Returns:</span></dt><dd>condition number of the matrix</dd></dl>
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<h4>getInverseConditionNumber</h4>
<pre>public&nbsp;double&nbsp;getInverseConditionNumber()</pre>
<div class="block">Computes the inverse of the condition number.
 In cases of rank deficiency, the <a href="../../../../../org/apache/commons/math3/linear/SingularValueDecomposition.html#getConditionNumber()"><code>condition
 number</code></a> will become undefined.</div>
<dl><dt><span class="strong">Returns:</span></dt><dd>the inverse of the condition number.</dd></dl>
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<h4>getRank</h4>
<pre>public&nbsp;int&nbsp;getRank()</pre>
<div class="block">Return the effective numerical matrix rank.
 <p>The effective numerical rank is the number of non-negligible
 singular values. The threshold used to identify non-negligible
 terms is max(m,n) &times; ulp(s<sub>1</sub>) where ulp(s<sub>1</sub>)
 is the least significant bit of the largest singular value.</p></div>
<dl><dt><span class="strong">Returns:</span></dt><dd>effective numerical matrix rank</dd></dl>
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<h4>getSolver</h4>
<pre>public&nbsp;<a href="../../../../../org/apache/commons/math3/linear/DecompositionSolver.html" title="interface in org.apache.commons.math3.linear">DecompositionSolver</a>&nbsp;getSolver()</pre>
<div class="block">Get a solver for finding the A &times; X = B solution in least square sense.</div>
<dl><dt><span class="strong">Returns:</span></dt><dd>a solver</dd></dl>
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